Why choose sets to be the primitive objects in mathematics rather than, say, tuples?

Sets are defined in such a way that $\{a,a\}$ is the same as $\{a\}$, and $\{a,b\}$ is the same as $\{b,a\}$. By contrast, the ordered pair $(a,a)$ is distinct from $(a)$, and $(a,b)$ is distinct from $(b,a)$.

Intuitively, it would seem useful to draw a distinction between two collections if they are ordered differently, or if one collection has a different number of copies of an element to the other. For instance, this would mean that the collection of prime factors of $6$ would be different to that of $12$. However, it is the set, rather than the tuple, that is chosen as the primitive object. Why is it useful for the foundations of mathematics that sets have very little "structure", and would their be any difficulties in choosing tuples to be the primitive object instead?

Solution 1:

The simpler a kind of object is, the more impressive its expressive power becomes. Sets are more compelling than tuples as an answer to the question, "How structurally simple can we make the foundations of mathematics?" Now that may not be a question one cares about - in particular, structural simplicity is often at odds with actual usability, and moreover one might reasonably reject simplicity as an inherent virtue in this context at all - but it is a meaningful question. And the early history of modern logic (especially around logicism) highlights this question.

So is there a problem with taking tuples - or even more complicated objects - as our "ground objects" for founding mathematics? Well, it depends on exactly what our desiderata of that foundational project are. Certainly there's no fatal obstacle, in the sense that from a technical perspective a tuple-based approach will be as safe and expressive as a set-based approach, but we might still have additional reasons to prefer the sets-based approach.

For what it's worth, I believe that simplicity-of-base-objects as a foundational criterion has lost a lot of steam over the last few decades, especially outside of the mathematical logic community: the notion of structural foundations of mathematics has grown in popularity.

Solution 2:

Sets can be infinite, and there are different cardinalities of infinite sets; we know by Cantor's diagonal argument that there are "more" real numbers than natural numbers, so not every set can be indexed by the natural numbers. Therefore, many sets including the set of real numbers simply cannot be represented as an infinitely long tuple.

There's some talk in the comments about how you could define tuples to be indexed by an arbitrary ordinals, but in that case it seems to me that infinite tuples would not be primitive objects at all; they would be defined in terms of other things (ordinals and functions).

Solution 3:

The mainly reason why set theory chose sets not natural mathematical objects (such as ordered pair, relations, functions, numbers and so on) as a base is: almost all the mathematical objects in modern mathematics can be reduced to sets, or the existence of almost all the mathematical objects can appeal to axioms of set theory for which $\mathsf{ZFC}$ is usually considered. And the reason why we pursuit such a general goal is because of the third mathematical crisis which is led by Russell paradox.

In fact, before the third mathematical crisis, set theory is developed as naive set theory and after that into axiomatic set theory. Naive set theory treat many natural mathematical objects as a base while axiomatic set theory chose sets.

Almost all the mathematicians just want to get a solid foundation to choose sets as a base in their thought which brings almost nothing to their mathematical practice because the sets as a base are too general, and so they still do what they do as usual in their mathematical practice.